During a lecture at UCLA, Serge Lang asked what is the most important function in mathematics. This question is quite personal and every person certainly has his own opinion. In fact, a professor spoke out loud that the constant function is the most important function.
Serge Lang suggested . The function is self-dual (i.e. its Fourier transform is itself, see below) and the property is essential to the functional equation of the Riemann zeta function.
For , the Fourier transformof is a function on defined by
Since , the above integral is absolutely convergent.
Theorem 1 The function is self-dual, i.e., .
Proof 1.We will use Cauchy’s theorem from the theory of complex analysis. For another proof without the theory of complex analysis, see proof 2 below.
We will show that the integral on the right hand side is and completes the proof. Let be the contour of the rectangle with vertices , , , . By Cauchy’s theorem
Let , the second term and the fourth term . Thus
The last equality is a standard application of double integration in polar coordinates.
Before we proceed to the second proof, we need the following lemma.
Lemma 2 Suppose .
(i)If has a continuous derivative and , then
(ii) Let . Suppose is integrable, then
In the last step we use the fact that and hence . This proves (i). Next Suppose is integrable. Then is also integrable. Hence
Thus (ii) follows.
Proof 2. Let . Simple calculation shows that
Applying the Fourier transform to the above equation, by the above lemme we obtain
Solving the ordinary differential equation, we obtain
where is a constant. To find , we set